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> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..." My kids are learning m

by lhorie 6y ago

> teaching students one thing, and then later coming back and saying "well, that thing you were taught before isn't actually correct..." My kids are learning multiplication. I honestly don't feel like they're being lied to, or confused about what they are being taught. Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). It's certainly not how I learned (I had to do rote memorization of the tables), but hey they can recite tables now too. Easing into multiplication via addition doesn't feel wrong IMHO. In fact, at later grades, I'd have to learn about associative/commutative properties anyways. My daughter in particular has a somewhat peculiar story: she is in kinder and learning multiplication from a game[0] that her older brother started playing, and the way they introduced multiplication is by telling her that to calculate the area of a rectangle, she needs to literally count the number of blocks that compose it. Is it "wrong"? I guess. Blocks aren't really proper units, but she gets that a rectangle made of 4x5 blocks has an area of 20 blocks, first by stumbling n' counting, then adding rows then eventually just memorizing the multiplication fact, so mission accomplished? Things build up from there, first without the grid, then into area of more complex polygons, spin off into word problems, etc. IMHO what helps kids learn is just repeatedly seeing where the memorized mechanisms can be applied and where they can't. Eventually multiplication becomes second nature and the addition/subtraction crutches come off. [0] https://www.prodigygame.com/ https://www.prodigygame.com/

com2kid 6y ago
> (e.g. `9 x N` is initially taught as `10 x N - N`) That isn't a crutch, that is how large numbers can be multiplied in one's head! 90 * 21 is (90 * 20) + 90 Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally. Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14. Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memorized and I'll add my way from there. I don't remember 7 * 6 but I know its 7 * 7 - 7.
- FPGAhacker 6y ago
  > Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14. Yeah, that’s interesting how we do things in our head like that. I would frequently break that down as 9*10 + 9*4 in my head. But I might also do it as 10*14 - 10 - 4. I’ll try different ways in my head until I can solve one of them easily.
jacobolus 6y ago
> Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system. Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying. Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together. 18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108 vs. 18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108 Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.
pdonis 6y ago
> I honestly don't feel like they're being lied to, or confused about what they are being taught. They might not be. If they're just being taught how to multiply in particular cases (whole numbers) using repeated addition, then they're not being lied to. If they're being told straight up that, while multiplication is not identical to repeated addition, they can ease into multiplication by learning repeated addition, even better. They're only being lied to if they're being told, authoritatively, that multiplication is repeated addition, no ifs, ands, or buts. And yes, I have had teachers like that, and I suspect many others have too.

by lhorie 6y ago

com2kid 6y ago

> (e.g. `9 x N` is initially taught as `10 x N - N`) That isn't a crutch, that is how large numbers can be multiplied in one's head! 90 * 21 is (90 * 20) + 90 Right after removing that extra 90, now you have 9 * 2 * 10 * 10 + 90, which is easy to do mentally. Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14. Did I have to memorize 0x0 through 12x12? Yup. But now days I have a few key mid points memorized and I'll add my way from there. I don't remember 7 * 6 but I know its 7 * 7 - 7.

FPGAhacker 6y ago
> Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14 - 14. Yeah, that’s interesting how we do things in our head like that. I would frequently break that down as 9*10 + 9*4 in my head. But I might also do it as 10*14 - 10 - 4. I’ll try different ways in my head until I can solve one of them easily.

jacobolus 6y ago

> Yes, they start with addition and subtraction as crutches (e.g. `9 x N` is initially taught as `10 x N - N`). This is not a "crutch", beyond the extent to which a decimal place-value system is a crutch. I would instead call it a broader and more fluent view of the number system. Describing e.g. 18 = 2·10 – 2·1 instead of 1·10 + 8·1 is a perfectly valid alternative representation which happens to often be more convenient when multiplying. Either way when we multiply we break each multiplicand into a sum, multiply the components from each combinatorially, and then add the results together. 18·6 = (1·10 + 8)6 = 10(1·6) + 8·6 = 60 + 48 = 108 vs. 18·6 = (2·10 – 2)6 = 10(2·6) – 2·6 = 120 – 12 = 108 Regularly discussing the alternative ways to represent a number and choosing the most convenient for the current goal builds what is called "number sense": fluency with the place-value system, basic properties of integers, relationships between numbers, base ten, and in a broader way facility with manipulating data structures.

pdonis 6y ago

> I honestly don't feel like they're being lied to, or confused about what they are being taught. They might not be. If they're just being taught how to multiply in particular cases (whole numbers) using repeated addition, then they're not being lied to. If they're being told straight up that, while multiplication is not identical to repeated addition, they can ease into multiplication by learning repeated addition, even better. They're only being lied to if they're being told, authoritatively, that multiplication is repeated addition, no ifs, ands, or buts. And yes, I have had teachers like that, and I suspect many others have too.